UV unwrapping, texture only appears on 5 sides of cube
First off you are using the blender internal. It is a little different for cycles.

You are in textured mode, textured mode uses the lights on the active layer(s) to shade the objects.

So you can either add lights to make a better lighting setup. 

Or switch the shading mode to Material. It will work like you are expecting, with your texture viable on all sides regardless of the lighting.